To find the length from point A to point B, we can follow these steps:

1. Identify the key elements and understand the geometric relationships:

   • The radius of the clock is $r = 15$ cm.
   • The distance from the top of the clock (point D) to the hanger (point B) is 10 cm.
   • Segments AB and BC are tangent to the circle at points A and C respectively.

2. Visualize the Geometry:

   • Since DE and EF are radii of the circle, both are 15 cm.
   • DF is the diameter, so $DF = 2r = 30$ cm.
   • Point D is directly above point E, and point B is vertically above point E by 10 cm.

3. Locate Point E:

   • The center of the clock is point E, and DE = EF = 15 cm.

4. Set up the coordinates:

   • Place point E at the origin (0,0) of a coordinate plane.
   • Therefore, point D is at (0, 15) because it is directly above E by one radius length.
   • Point B is directly above E by 10 cm, so point B is at (0, 25).

5. Identify the tangents:

   • Points A and C are the points of tangency.
   • The tangents from a common external point (B) to a circle are equal in length.

6. Right triangle consideration:

   • Right triangle formed by A, B, and E (A tangent from B and A is a right angle).
   • Let $AB = x$ and $BE = 15$.

7. Use the Pythagorean Theorem: $AB^2 + BE^2 = BD^2$ Where $BD = 10$ cm.

8. Solve for AB: $AB^2 + 15^2 = 25^2$ $AB^2 + 225 = 625$ $AB^2 = 400$ $AB = \sqrt{400} = 20 \text{ cm}$

Thus, the length from point A to point B is $\boxed{20 \text{ cm}}$.

Do you want more details or have any questions?

Here are some related questions you might find interesting:

1. How do you determine the length from point C to point B?
2. What is the total length of the wire from A to C passing through B?
3. How do you find the angle between the tangents AB and BC?
4. How can you determine the distance from point E to the tangent points A and C?
5. What would happen if the radius of the clock was different?
6. How do you determine the coordinates of points A and C?
7. How does the vertical distance from point D to B affect the tangent lengths?
8. How do you calculate the lengths if the tangents intersect at a different point than B?

Tip: When dealing with tangent lines to a circle, remember that tangents from a common external point to the circle are always equal in length.